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11 Short Run Production Function
14.01 Principles of Microeconomics, Fall 2007

Chia-Hui Chen

October 1, 2007

Lecture 11
Production Functions
Outline
1. Chap 6: Short Run Production Function
2. Chap 6: Long Run Production Function
3. Chap 6: Returns to Scale
1 Short Run Production Function
In the short run, the capital input is fixed, so we only need to consider the
change of labor. Therefore, the production function
q = F(K, L)
has only one variable L (see Figure 1).
Average Product of Labor.
Output q
APL = = .
Labor Input L
Slope from the origin to (L,q).
Marginal Product of Labor.
∂Output ∂q
MPL = = .
∂Labor Input ∂L
Additional output produced by an additional unit of labor.
Some properties about AP and MP (see Figure 2).
• When
MP = 0,
Output is maximized.
• When
MP > AP,
AP is increasing.
10

9

8

7

0 1 2 3 4 5 6 7 8 9 10

L

0
1
2
3
4
5
6
AP
MP
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21 Short Run Production Function
0 1 2 3 4	 5 6 7 8 9 10

L

0
5
10
15
20
25
30
35
40
Q
Figure 1: Short Run Production Function.
Figure 2: Average Product of Labor and Marginal Product of Labor.
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32 Long Run Production Function
• When
MP < AP,

AP is decreasing.

•	 When

MP = AP,

AP is maximized.
To prove this, maximize AP by first order condition:
∂ q(L)
= 0
∂L L
=⇒
∂q 1 q
− = 0
∂L L L2
=⇒
∂q q
=
∂L L
=⇒
MP = AP.
Example (Chair Production.). Note that here APL and MPL are not continuous,
so the condition for maximizing APL we derived above does not
apply.
Number of Workers Number of Chairs Produced
0 0
1 2
2 8
3 9
APL MPL
N/A N/A
2 2
4 6
3 1
Table 1: Relation between Chair Production and Labor.
2 Long Run Production Function
Two variable inputs in long run (see Figure 3).
Isoquants. Curves showing all possible combinations of inputs that yield the
same output (see Figure 4).
Marginal Rate of Technical Substitution (MRT S). Slope of Isoquants.
dK
MRTS = −
dL
How many units of K can be reduced to keep Q constant when we increase
L by one unit. Like MRS, we also have
MPL
MRTS = .
MPk
4
Q
10
9
8
7	 k=3
6
5	 k=2
4	 k=1
3
2
1
0
0 1 2 3 4	 5 6 7 8 9 10
L
k	
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
L
2 Long Run Production Function
Figure 3: Long Run Production Function.
Figure 4: K vs L, Isoquant Curve.
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k
10

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5

4

3

2

1

0

0 1 2 3 4	 5 6 7 8 9 10
L
53 Returns to Scale
Proof. Since K is a function of L on the isoquant curve,
q(K(L), L) = 0
=⇒
∂q dK ∂q
+ = 0
∂L dL ∂L
=⇒
dK MPL
− = .
dL MPK
Perfect Substitutes (Inputs). (see Figure 5)
Figure 5: Isoquant Curve, Perfect Substitutes.
Perfect Complements (Inputs). (see Figure 6)
3 Returns to Scale
Marginal Product of Capital.
∂q(K, L)
MPK =
∂K
Marginal Product of Labor

K constant , L ↑ → q?

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6
k
10

9

8

7

6

5

4

3

2

1

0

0 1 2 3 4	 5 6 7 8 9 10
L
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3 Returns to Scale
Figure 6: Isoquant Curve, Perfect Complements.
Marginal Product of Capital
L constant , K ↑ → q?
What happens to q when both inputs are increased?
K ↑ , L ↑ → q?
Increasing Returns to Scale.
•	 A production function is said to have increasing returns to scale if
Q(2K, 2L) > 2Q(K, L),
or
Q(aK, aL) = 2Q(K, L), a < 2.
• One big firm is more efficient than many small firms.
• Isoquants get closer as we move away from the origin (see Figure 7).
7
k
10

9

8 Q=3
7

6

5 Q=2
4

3

2 Q=1
1

0
0 2 4 6 8 10 12 14

L

3 Returns to Scale
Figure 7: Isoquant Curves, Increasing Returns to Scale.
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